Lie Symmetry Analysis, Traveling Wave Solutions, and Conservation Laws to the (3 + 1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation

In this paper, the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili(BKP) equation is studied applying Lie symmetry analysis. We apply the Lie symmetry method to the (3 + 1)-dimensional generalized BKP equation and derive its symmetry reductions. Based on these symmetry reductions, some exact traveling wave solutions are obtained by using the tanh method and Kudryashov method. Finally, the conservation laws to the (3 + 1)-dimensional generalized BKP equation are presented by invoking the multiplier method.


Introduction
Nonlinear partial differential equations (NPDEs) are widely used as models to describe a great number of complex nonlinear phenomena which appear in many fields, such as hydrodynamics, biology, plasma physics, fluid dynamics, solid state physics, optics, and applied mathematics. To really understand such phenomena describing in nature, searching for exact solutions of NPDEs plays an important role in the study of nonlinear science. In recent years, more and more methods have been proposed, such as the inverse scattering method [1], the Bäcklund transform method [2], the Darboux transform method [3], the Hirota bilinear transformation method [4], the Exp-function method [5], the tanh-function method [6], the sine-Gordon equation expansion technique [7], and the Kudryashov method [8].
e Lie symmetry method presented by Lie [9] is one of the well-known methods for obtaining exact solutions of nonlinear PDEs. Up to now, the Lie symmetry method has been applied to a number of mathematical and physical models, see [10][11][12][13][14] and references therein. is method is effective to get similarity solutions and solitary wave solutions of NPDEs. e study of different BKP equations has attracted a considerable size of research. Different forms of BKP equations were studied by using the Hirota method, the multiple exp-function algorithm, the Pfaffian technique, the Wronskian technique, and the Bäcklund transformation [15][16][17][18][19][20][21]. In this paper, by means of the Lie symmetry group method, we consider the following (3 + 1)-dimensional generalized BKP equation: where u is a real differentiable function of the scaled spatial coordinates x, y, and z and temporal coordinate t, while the subscripts denote the partial derivatives. is equation was first presented by Ma et al. [15]. When z � 0, the (3 + 1)dimensional generalized BKP equation can be reduced to the following (2 + 1)-dimensional BKP equation: Equation (1) has been investigated by different methods. Following the linear superposition principle of exponential waves, Ma et al. obtained an N-wave solution for equation (1) in [15]. A bilinear Bäcklund transformation and a class of exact Pfaffian solutions for equation (1) were established using the Pfaffian technique and Hirotas bilinear operator identities in [16]. Moreover, Wazwaz [17] established two sets of distinct kinds of multiple-soliton solutions under specific conditions for equation (1) using the simplified form of the Hirota method. Soliton solutions in Wronskian form of (1) were also presented in [18,19]. Multiple wave solutions and auto-Bäcklund transformation for the (1) were obtained in [20]. e lump solutions, periodic waves, and rogue waves as well as interaction solutions of (1) were obtained in [21].
As far as we know, the Lie symmetry analysis and conservation laws to the (3 + 1)-dimensional generalized BKP equation (1) have not been discussed. e main purpose of this paper is to study the Lie symmetry analysis method [22][23][24], exact traveling wave solutions, and conservation laws of equation (1). e rest paper is arranged as follows: Section 2 is devoted to describe the Lie symmetry vectors and symmetry reductions using Lie symmetry analysis. In Section 3, various exact traveling wave solutions are attained after reductions process by using the tanh method and Kudryashov method. Section 4 is devoted to find the conservation laws of equation (1) by utilizing the multiplier method. Some conclusions are made in Section 5.

Exact Solutions of Equation (19) Using Tanh Method.
In this subsection, we apply the tanh method to obtain solutions of equation (19). Suppose that the solution of equation (19) can be expressed as where n is a positive integer and a i (i � 0, 1, 2, . . . , n) are all constants. Z � tanh(ξ) is a new independent variable, the derivatives of U with respect to ξ can be written as Balancing the linear term of the highest order with the highest order nonlinear term, we can easily obtain n � 1 and thus (20) becomes

Complexity
Substituting (21) into (19) and using (22), we have an algebraic equation, which on splitting with respect to the powers of Z gives the following system: whose solution is a 0 is an arbitrary constant, Substituting (24) into (22), we obtain the traveling wave solution to equation (19) us, the traveling wave solution of the (3 + 1)-dimensional generalized BKP equation (1) is where a 0 , α, β, and ω are constants and satisfy the algebraic
For the multiplier Λ 2 � 6F 1 (t) + F 1 ′ (t)x, we obtain the conservation law of equation (1) as with the following components: For the multiplier Λ 3 � F 2 (ξ), the conservation law of equation (1) is and the components are

Conclusion
e exact solutions of the (3 + 1)-dimensional generalized BKP equation (1) have been constructed using different methods by many authors.
e Pfaffian technique, the Hirotas bilinear method, and the bilinear Bäcklund transformation were applied by many researchers, and various exact solutions such as Pfaffian solutions, lump wave solutions, periodic wave solutions, rogue wave solutions, and interaction solutions were derived. Based on the Hirota bilinear form, the Wronskian method, and the multiple Expfunction method, the Grammian and Pfaffian techniques were used to construct multiple wave solutions to the (3 + 1)dimensional generalized BKP equation. Different from these existing results, we choose another approach to study equation (1). In this paper, the combination of Lie symmetry method and the symbolic computation is applied to equation (1). e geometric vector fields of this equation are presented for the first time in the literature. With the aid of Lie symmetry reductions, some exact traveling wave solutions are obtained by using the tanh method and Kudryashov method to the reduced equation of (1). e conservation laws of equation (1) are presented by using the multiplier method at the end of the paper. We can find that these are different approaches to find exact solutions of the (3 + 1)-dimensional generalized BKP equation, and we hope that some more interesting solutions of equation (1) are shown in the future.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.